HairBSDF, sample_f, cosTheta

Average Error: 0.2 → 0.2

Time: 14.0s

Precision: binary32

Cost: 6880

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

↓

\[1 + v \cdot \log \left(u + \frac{e^{\frac{-2}{v}}}{\frac{1}{1 - u}}\right) \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

↓

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (/ (exp (/ -2.0 v)) (/ 1.0 (- 1.0 u))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

↓

float code(float u, float v) {
	return 1.0f + (v * logf((u + (expf((-2.0f / v)) / (1.0f / (1.0f - u))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

↓

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + (exp(((-2.0e0) / v)) / (1.0e0 / (1.0e0 - u))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

↓

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) / Float32(Float32(1.0) / Float32(Float32(1.0) - u)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

↓

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + (exp((single(-2.0) / v)) / (single(1.0) / (single(1.0) - u))))));
end

Derivation

1. Initial program 0.2

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

2. Applied egg-rr 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\frac{e^{\frac{-2}{v}}}{\frac{1}{1 - u}}}\right) \]

3. Final simplification 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \frac{e^{\frac{-2}{v}}}{\frac{1}{1 - u}}\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	6816

\[1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \]

Alternative 2
Error	0.3
Cost	6752

\[1 + v \cdot \mathsf{log1p}\left(\left(1 - u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \]

Alternative 3
Error	1.2
Cost	6688

\[1 + v \cdot \log \left(u + e^{\frac{-2}{v}}\right) \]

Alternative 4
Error	3.0
Cost	4036

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot {\left(1 - u\right)}^{2}}{v}\right)\\ \end{array} \]

Alternative 5
Error	2.9
Cost	3684

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	30.2
Cost	32

\[-1 \]

Alternative 7
Error	4.1
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))